Arthur Miller and his daughter, Rebecca, in 1995. She approved the current hit revival of “Death of a Salesman” on Broadway, but it likely won’t be seen again soon. Photo: Lynn Goldsmith/Corbis

Shortly before he died in 2005, Arthur Miller smiled at his daughter, Rebecca, and said, “Catch!”

What he threw her way was nothing less than his literary legacy, a body of work that includes some of the most celebrated plays of the 20th century — “All My Sons,” “The Crucible,” “Incident at Vichy,” “A View From the Bridge” and, of course, “Death of a Salesman.”

Judging from the success of the Broadway revival of “Salesman,” Rebecca Miller hasn’t dropped the ball.

Directed by Mike Nichols and starring Philip Seymour Hoffman, “Salesman” is playing to sold-out houses at the Barrymore and posting weekly grosses of nearly $1 million.

The production is likely to pick up several Tony Awards next month, including Best Revival.

The limited run ends Saturday, so you might want to beg, borrow or steal a ticket, since another “Salesman” is unlikely to come this way for a long time.

“Unless there is a compelling reason to say yes, my basic attitude is to say no,” Rebecca Miller says, summing up her approach to her father’s literary estate.

Speaking from her home in Connecticut, the 49-year-old says, “I want the plays to remain special and be framed in such a way that they don’t become Monet placemats. My father wasn’t smug about his future life as an author. He realized that nobody knows what will happen to their work. What might seem inevitable is not. Even the most popular writers fall out of favor. So you need to care for the plays.

“In many ways, I’m much pickier than he was. When he was alive, he said yes to a lot [of productions]. After all, he could always create more plays. I have to be much more protective.”

She’s also reluctant to let her role as literary cop interfere with her own work as a screenwriter and director, whose movies include “Angela” and “The Private Lives of Pippa Lee.”

“My father’s desire was that I not become enslaved by it all. If I had to attend every production around the world, I’d never have time to do my own writing. So I’m careful not to let it eclipse my own career.”

Miller did set some precedents that have guided his daughter.

For one thing, cutting. Don’t even think about it.

“There was a theater group, a very good one, that wanted to do ‘The Crucible’ in double time, cutting half of it,” she recalls. “Dad said, ‘Write your own play.’ ”

Worldwide, “The Crucible,” set in witch-hunting Salem, is Miller’s most popular play. It was also made into a fine movie starring Rebecca’s husband, Daniel Day-Lewis. But she keeps a tight leash on the play because “people always want to mess around with it. The language is more difficult, more arcane. People want to make it more comprehensible. I say, ‘No.’ ”

Of Miller’s best-known plays, the only one that hasn’t been revived on Broadway is “Incident at Vichy,” a compelling World War II drama about a group of Frenchmen who’ve been rounded up by the Gestapo. She’d love to see a production of it in New York at some point.

She’s also fond of some of his more obscure plays, such as “A Memory of Two Mondays” and “The Last Yankee.”

“They’re wonderful gems, but it’s hard to find a place for them,” she says. “On Broadway you have to have huge, heavy stars. It’s got to be a big night out. These plays are meditations. They don’t fulfill those kinds of expectations.

As for “After the Fall” and “Finishing the Picture” — Miller’s two plays about his second wife, Marilyn Monroe — Rebecca remains wary. (She’s the daughter of Miller’s third wife, the photographer Inge Morath.)

“Anything that’s got the shadow of Marilyn in it — even something that has just a slight taste of her — gets overshadowed by her,” she says.

“Finishing the Picture,” Miller’s final play, is about the making of Monroe’s last movie, “The Misfits,” for which he wrote the screenplay. There has been just one production, at Chicago’s Goodman Theatre in 2004.

It’s a fascinating story with some terrific parts for larger-than-life stage actors. “But I’m holding back on it,” says Rebecca. “I’m guarding it. Sometimes it’s good to hold.”

She calls “After the Fall,” which Miller wrote in 1963, shortly after Monroe died, “a wonderful play that unfortunately in his time got completely read as an autobiographical work about her. You can’t pretend she’s not there, but at the same time it is about other things. There are meditations on the Holocaust and how we all have murder inside us.”

Rebecca says she’d like to see a production that “skews the play” away from the Monroe character.